Showing papers on "Piecewise published in 1968"

Finite State Continuous Time Markov Decision Processes with a Finite Planning Horizon

Abstract: The system we consider may be in one of n states at any point in time and its probability law is a Markov process which depends on the policy (control) chosen. The return to the system over a given planning horizon is the integral (over that horizon) of a return rate which depends on both the policy and the sample path of the process. Our objective is to find a policy which maximizes the expected return over the given planning horizon. A necessary and sufficient condition for optimality is obtained, and a constructive proof is given that there is a piecewise constant policy which is optimal. A bound on the number of switches (points where the piecewise constant policy jumps) is obtained for the case where there are two states.

On the design of linear systems with piecewise-constant feedback gains

Abstract: This paper presents the theory and the computational algorithm required to determine the piecewise-constant feedback gains for the optimal control of a linear system with quadratic performance index over a finite control interval [t_{0} T] . A third order example illustrates the relation of the piecewise-constant gains to the truly optimal time-varying gains.

Variational and optimal control problems with delayed argument

Abstract: This paper deals with variational and optimal control problems with delayed argument and presents analogs of the classical necessary conditions for optimality for problems in (n + 1)-space. It is mainly concerned with the functional $$J(y) = \int_a^b {f[t,y(t\user2{--}\tau ),y(t),\dot y(t\user2{--}\tau ),\dot y(t)] dt} $$ There are no side conditions; τ is a positive real number; andy is a continuous piecewise smooth vector function havingn components. The fundamental lemma of the calculus of variations is used in deriving an analog of the Euler equations. The usual construction is utilized in obtaining analogs of the Weierstrass and Legendre conditions. Also found is a fourth necessary condition involving the least proper value associated with a boundary value problem related to the second variation. A sufficient condition is obtained by the use of a simple expansion method. The last station of the paper outlines an extension of a maximal principle obtained by Hestenes to control problems which involve delays in both the state variable and the control variable.

Analytic solutions of limit cycles in a feedback-regulated converter system with hysteresis

Abstract: A mathematical model is derived for the voltage step-down dc-to-dc converter in which a hysteretic bistable trigger circuit is used to regulate the output voltage. Normalized second-order differential equations are derived for the output-voltage error, or output-voltage ripple, measured with respect to a constant reference. The method of successor functions is applied to the piecewise analytic phase plane trajectory for the errors and the conditions leading to a limit cycle are initially formulated in two transcendental equations. With the objective of obtaining an analytic solution for the period of the limit cycles these two equations are then replaced by approximate algebraic equations which are solvable in general terms. The approximations are carefully based on properties that are typical of all converters of this type, and lead to quite simple but accurate expressions for the period of the limit cycle in terms of arbitrary system parameters. Expressions for the amplitude of the limit cycle are also given, and its stability is tested. A numerical example based on an actual representive system is given. Certain unusual characteristics of the limit cycle as a function of certain system parameters are pointed out.

Minimum-weight design with piecewise constant specific stiffness

Abstract: Much of the literature on minimum-weight design is concerned with structures with continuously varying dimensions. Structures of this kind achieve an absolute minimum of weight and thereby provide a useful standard of comparison, but their manufacturing cost is prohibitive in most applications. As a step toward the establishment of optimality conditions for more practical structures, the present paper discusses optimal design of sandwich beams and frames with segmentwise constant cross sections. Optimality is shown to require that the mean square stress in the face sheets has the same value for all segments. The use of the optimality condition is illustrated by examples.

The Piecewise Solution of the Impedance Matrix Load Flow

Abstract: This paper presents a method for solution of the load flow problem which involves tearing a power system into pieces. This permits use of the impedance matrix method on larger systems than was previously feasible. A detailed tearing algorithm is given. Its validity is examined by presentation of test results for a small sample system. The new method retains the same features and convergence characteristics as the original impedance matrix method [1].

Flexible bars subjected to arbitrary discrete loads and boundary conditions

Abstract: A method of analysis for the large deflections of piecewise prismatic elastic bars subjected to an arbitrary number of discrete loads is presented. By combining the general solution of the nonlinear differential equation of bending and the equilibrium equations for each of the intervals between the load points and points of discontinuity in the flexural rigidity, along with the given boundary conditions, the problem is formulated as a system of simultaneous nonlinear equations which is solved by a modified Newton-Raphson iteration procedure. Four examples are given to illustrate the application of the method for the treatment of beam-columns with various loading and boundary conditions.

Identification of distributed parameter systems by stochastic approximation

Abstract: Stochastic approximation schemes are applied to the identification of linear distributed parameter systems The dependent function is assumed piecewise continuous and is approximated by a finite number of functions chosen from a "complete" system of orthonormal functions It is also assumed that noisy measurements of the function are available at random points in space and time Therefore, the identification is performed off-line The constant parameters multiplying the approximating functions are obtained sequentially by stochastic approximation algorithms that minimize a mean-square error performance criterion

Exponential approximation with piecewise linear error criteria

Abstract: The problem of approximating a continuous function on the interval [0, T] with a linear combination of real exponential functions is considered. Rather than restricting the analysis to mathematically well behaved error criteria, such as least-squares, the theory of distributions is used to analyze criteria based on piecewise linear weighting of the error. For such criteria, it is found that the first and second derivatives of the error measure E with respect to the parameters of the approximating function can be obtained from exact and easily evaluated formulae. A descent search to find the solution to the approximation problem by minimizing E is thus economically feasible, since no inaccurate and time-consuming numerical integrations are required to produce the needed derivatives. As an example of such a criterion, approximation in the L1 norm is treated.

Synthesis of piecewise linear characteristics

Abstract: An analytical method of synthesizing two or more non-linear characteristics put in parallel or in series, is described in the present study. The method is based on set theory and is restricted to n...

Dual Analysis of a Multiweb Sweptback Wing Model

Abstract: THE idea of a dual analysis in finite elements of a given structure was put forward in Ref. 4. The first analysis should be of the displacement type, using conforming displacement models of the finite elements. This results in a continuous, piecewise differcntiable displacement field in the whole structure, for which linear elasticity theory predicts lower bounds to the local static influence coefficients. The second analysis should be based on equilibrium models of the finite elements. The stress field within the structure is then continuously transmitted across the interfaces and satisfies detailed equilibrium conditions in the interior of each element. This property furnishes upper bounds to the influence coefficients.

On the Design of L,inear Systems with Piecewise-Constant Feedback Gains

Abstract: system is generated by a feedback law of the form u*(t) = -L*(t)x(t). The elements of the matrix L*(t) are the time-varying feedback gains which are computed from the solution of a nonlinear matrix Riccati differential equation. The design approach taken in this paper is to replace the optimal gains L*(t) with a set of piecewise-constant gains which are chosen in an optimal fashion. The resulting piecewise-constant gains are thus relatively easy to implement; it is shown that they can' approximate the optimal gains arbitrarily closely. A convergent algorithm is presented for computing these gains. Lastly, a suboptimal control law is calculated for a third order example. The research reported below rests heavily on the theoretical results of Kleinman and Athans [6] in which the general problem of control with fixed time-varying structures was investigated. A suboptimal linear regulator problem was precisely formulated and necessaqconditions for its solution were derived. In this paper, the pertinent results of [6] will be summarized.

The scattering of plane waves by a piecewise-uniform plasma with a zero layer

Abstract: The p rob lem of propagat ing e l ec t romagne t i c waves along a uniform p l a s m a waveguide is the subject of cons iderable study today; d i spe r s ion equat ions have been found for waves propagated along such guides, and the f ield maps of these waves have been i nves t i gated. In the case of a bounded p l a sma waveguide (or unbounded waveguide with jump changes in densi ty along its axis) the ex is tence of d iscont inui t ies leads to the appearance of new effects of mutual convers ion and sca t t e r ing of the natura l waveguide modes, the study of which is of g rea t impor tance in p rac t i ca l appl icat ions . In the genera l case the analyt ical study of the p r o b l em of the d i f f rac t ion of e l e c t romagne t i c waves in a p la sma waveguide with jump changes in densi ty is compl ica ted by the need to solve Maxwel l ' s equat ions subject to the conditions of continuity of the tangential components of the f ie lds at the jumps in p l a sma densi ty . In this paper we shall cons ider the s ca t t e r i ng of TM waves subject to the condition that the un i form s e c t ions of the p l a sma are sepa ra t ed by an infini tely thin magnet ic s c r e e n {the tangential component of a m a g net ic f ield tends to ze ro at such a sc reen) . It was shown in [1] that an i so t rop ic p l a sma layer with r e sonance densi ty of the p la sma (Wp = w, where Wp is the p l a sma f requency of the l ayer , and w is the o p e r a ting frequency) has the p rope r t i e s of a magnet ic s c r e e n for TM waves, however thin it is (zero layer) . A un i f o r m resonance (Wp = w) l ayer of an an i so t rop ic p l a sma has s i m i l a r p rope r t i e s when a s t rong ex te rna l m a g net ic f ie ld w H >> Wp is d i r ec ted pe rpend icu la r to the plane of the l aye r (see Appendix). (If the ex te rna l f ield is d i r ec ted along the plane of the l ayer , the r e f l ec t ive p rope r t i e s of the l aye r for p l a sma resonance {w 2 = = ~ + o9~) depend es sen t i a l ly on the angle of inc id ence, as shown in [2].) It cart b e s h o w n t h a t t h e r e f l e c t i r e capaci ty of a l ayer is a lso re ta ined in the absence of s t r i c t r esonance , for angles of incidence not too near 7r/2, if the condition

Solution of large power system networks by Diakoptics

Abstract: This paper outlines the results of experience with the application of the piecewise solution of large networks using the method of Diakoptics of Kron ( 1 ); the problems encountered in the application of the method in its present formulation; and the areas indicated for further study and development.